Lets imagine 2 cubes, CA and CB, that share a face. On CA lives Adam and on CB lives Ben. Adam and Ben are perfectly flat. Under no conditions can Adam leave the surface of CA; ditto for Ben and CB. Because the cubes share a face, is it possible for Adam and Ben to meet? If so, what does Adam see when Ben travels from the shared face to a different face?

A cube is already not a nicely behaved manifold (because it has curvature singularities at the corners.

A better behaved case would be two spheres sharing a point. The answer still isn't really possible to answer well because actual physics restricts the physics to all take place on one manifold. In the case of the two spheres touching, we aren't on a manifold any more; we're in some space with less structure which just has two subspaces which are manifolds.

Basically, we can define the behaviour at the point of contact to be however we want. The easiest choice is to say everything stays in its original manifold but this is boring because then no-one can talk to anyone else.

The next easiest choice is to require everything goes to the other manifold if it goes through the point of contact. In order to keep from having singularity in the curvature's derivative, both spheres must have the same radius but we still a freedom to choose the sense of the curvature because this is an extrinsic property. Essentially we can choose whether the two spheres coincide everywhere or don't coincide except at the point of contact. In the first case, no interesting physics happens particularly, you just jump from one manifold to another if you go through the point of contact. If the spheres coincide nowhere except at the point of contact, you'd get some interesting physics happening because a right-handed set of vectors on one manifold would be left-handed once it transfers to the other manifold.

If you want to share a region between the two manifolds, you run into bigger issues; in particular, consider something like two bubbles touching

The two simplest possibilities are illustrated by thinking about an ant walking of the inside or the outside of a bubble. For an ant on the inside, the ants will simply stay within their own bubble whilst an ant on the outside will see a region of their world suddenly absent and replaced with a much larger region: the second bubble.

In order to have more interesting behaviours you need the ant to be able to cross through the surface of the bubble at which point the interesting behaviour essentially boils down to the rules you posit for that crossing.

The cube situation (ignoring curvature singularities) should be similar to the touching bubbles.

@eSOANEMAre you assuming in all cases that Adam and Ben (what happened to Alice and Bob? Did they die? Are they stuck on a 0-manifold?) are point-like?What would change if they are 2-dimensional shapes? E.g. in the case of the two sphere sharing a point, they couldn't hop to the other sphere without tearing themselves apart.

eSOANEM wrote:In order to keep from having singularity in the curvature's derivative, both spheres must have the same radius

Could you expand on that? My intuition says the surfaces would line up perfectly regardless of the radii of the spheres.

Zohar wrote:You haven't provided us with enough information. Do Adam and Ben have sensory organs that allow them to watch things?

Adam and Ben have the 2 dimensional equivalent of all the senses humans have.

Are the cubes transparent?

Adam and Ben cannot turn their heads to look at the cube, so does it matter?The cubes have so much opaque things on their surface that an observer that lives in 3 spacial dimensions would describe them as opaque.

eSOANEM wrote:In order to have more interesting behaviours you need the ant to be able to cross through the surface of the bubble at which point the interesting behaviour essentially boils down to the rules you posit for that crossing.

An ant on a Klein bottle can travel from the 'inner' surface to the 'outer' surface without having to drill a hole, so what if the bubbles were Klein bottles instead of spheres?

Flumble wrote:↶@eSOANEMAre you assuming in all cases that Adam and Ben (what happened to Alice and Bob? Did they die? Are they stuck on a 0-manifold?) are point-like?What would change if they are 2-dimensional shapes? E.g. in the case of the two sphere sharing a point, they couldn't hop to the other sphere without tearing themselves apart.

eSOANEM wrote:In order to keep from having singularity in the curvature's derivative, both spheres must have the same radius

Could you expand on that? My intuition says the surfaces would line up perfectly regardless of the radii of the spheres.

I am indeed assuming Alice and Bob are pointlike. Larger people would have major issues in the touching-at-a-point case and probably end up dead.

What I mean with the curvatures is that each sphere will have a different (constant) curvature if the radii are different but, as we know from GR, curvature <=> gravity so if the spheres have different radii you'd have an imbalance of 'gravitational force'; ultimately you'd totally be able to set it up initially with different curvatures but there'd be some sort of runaway process and the only two possible endpoints without a discontinuity in the curvature is if they end up at the same radius or if one shrinks away to nothing.

jewish_scientist wrote:

eSOANEM wrote:In order to have more interesting behaviours you need the ant to be able to cross through the surface of the bubble at which point the interesting behaviour essentially boils down to the rules you posit for that crossing.

An ant on a Klein bottle can travel from the 'inner' surface to the 'outer' surface without having to drill a hole, so what if the bubbles were Klein bottles instead of spheres?

The behaviour on a Klein bottle isn't that much more interesting. The first time you go past the join you would go to the other manifold and come back but then the second time you see the bit that used to be where the other manifold was last time you looked.

Or, well, you could get those the other way round. Or you could go to the other manifold, keep going until you get back to the overlap region but see the other manifold's original region and think that you're trapped, going round again though you'd get back to your first manifold.

So there's still not much interesting physics, just some combination of the cases I outlined in my last post in various orders.

Zohar wrote:If these bubbles are opaque and they live on their surfaces (i.e. not part of/displacing the surface but sort of lying on top of it), why would there be any interaction at all?

Is there a difference between being on a manifold and being in a manifold?

eSOANEM, can you explain that again. It is getting a little too abstract for me to follow. How about we say that the first manifold is covered by/ contains a grassy field in the area that overlaps and a little beyond that; ditto for the second one, but with a ice. Would at some point Adam see a grassy field covered with frost? If not, then what would he see at the shared region?

(I wrote this last Monday; had to save as draft as power plummeted and a Ninja poster interupted my own hasty attempt at submission. Thanks for bringing it back to the top so I remembered to load the draft in again... May have to revisit to expand/reconsider, as promised, though.)

Going by the basic 'rules' of Flatland, but (here I'm going to assume a pair of ellipsoidoids with co-(but-anti-)planar touching faces1), I would presume there'd be a mysterious zone where 'others' would phase into existence at one edge and out again at the other edge. Assuming it is a fully-interacting co-existence rather than a ghostly apparition (which is itself an interaction; choose your own set of forces/energies to allow to exchange or not... Dark Matter, or the opposite?), it'd be interesting to know if synchronised convergence upon the zone could interlace the two membrane-residents. Or indeed anything else.

One problem with nuclear fusion is forcing the particles involved together, because of repulsive forces. If the forces travelled only along the relevent membrane, then a particle upon the interface might feel the force from the opposing membrane just the other side of the pinch-point, but its own repulsion may not propagate outside of the shared space onto the counterpart, to stop that other. Careful (or lucky/unlucky) synchronistion of entry into the shared zone might 'effortlessly' bring both particles together. Intended, that could solve problems; unintended, it may cause them. Also consider (say) two buckball cages interlocking in interesting ways (probably reconfiguring, as the bonds 'normalise' to the new situation) if directed inwards to the zone interface.

Assuming that there's not a phased interface (which could be experimentally determined), rather than instantaneous.

And also assuming that, given that the membranes are 'oppositely facing', certain properties (presumably universally determined by the direction of 'inside' and 'outside' of the membrane's bubble, in any system with such a sensible metadimension such as that) may be opposed. Which qualities? Take your pick. Or experimentally determine them. (If such a membrane-touch, or even cross-over2, exists in our own universe (within the observable universe, at least), there is exactly as little evidence that it creates 'anti-matter' by such opposition as there is that there are 'our universe' boundaries between matter-zones and anti-matter zones. Not that it doesn't rule out that our matter-heavy universe is (specifically) not anihilating with the anti-matter opposing 'verse due to the opposition rule making both the same to each other.)

Oh, bloomin' battery... Back later. I really need to get a charger sorted, out here in the wilds. This thought experiment is yet far from complete, and really could do with better editing down before posting, too, but it looks like I won't get the chance this side of midnight.

1 The alternative might be convergently curving surfaces from eccentrically aligned inner and outer 'bubbles' with an externally shared boundary of a compromise curvature based upon whatever limit of distortion you wish to allow (as with the alternative) to prevent a ring of curvature-singularity defining the meeting/seperation zone... But I like the Left-Handed/Right-Handed meeting idea more, and it disposes of tricky hierarchical bias between manifolds (or membrane-crosses-membrane interfaces elsewhere, where additional 'interactions' are possible, unless there's yet at least two additional dimensions beyond the brane, to possibly allow for Brunnian/Borromean overlap, beyond the contact point of contention... Am I waffling? I think I'm waffling. BYGTI),

2 Possibly a different beast, altogether. Consider the Star Trek "Nexus" ribbon, thingummy-cum-ploty-device as a representation. Given perpendicular components to the brane-on-brane action, rather than opposing natures (e.g. reversed arrows of time), there are imaginary relationships, for which space and time might 'usefully' interact, and cause whatever weirdnesses you would wish upon the scenario you're concocting.

Zohar wrote:If these bubbles are opaque and they live on their surfaces (i.e. not part of/displacing the surface but sort of lying on top of it), why would there be any interaction at all?

Is there a difference between being on a manifold and being in a manifold?

eSOANEM, can you explain that again. It is getting a little too abstract for me to follow. How about we say that the first manifold is covered by/ contains a grassy field in the area that overlaps and a little beyond that; ditto for the second one, but with a ice. Would at some point Adam see a grassy field covered with frost? If not, then what would he see at the shared region?

With an actual manifold, there is no difference between being in or on a manifold (or, well, I suppose I'd say that "in" is a thing points are and "on" is a thing people, fields etc are). When trying to argue what natural behaviour for two touching manifolds is though, it matters. Look back at the touching bubbles are think about a bug on the inside and another on the outside of the bubble. They'll each experience the transition zone differently.

With the shared region, it very much depends. You could posit all sorts of different behaviours because gr and our other theories make no claims about touching manifolds, only about individual manifolds and fields defined thereon.

Going back to the bubble example, the most natural 'shared region' to me seems to be one with no interactions (two bugs on the inside of each bubble cannot do any measurement to perceive the other). If you want interactions things get a lot more complicated and you have a lot more freedom to choose whatever you want but, you also need to think about how this structure came to be, I.e. what happens when two manifolds first touch, and make sure that you behaviour in this case is sensible.